Angle & Tolerance
The Frequency vs. Temperature characteristic of a quartz crystal is
determined by the angle at which the quartz plate is cut from the
quartz bar as referenced to the "Theta Angle". The cutting process
becomes a simple geometry problem to identify the angle of cut,
however the degree of accuracy is very challenging. Angles are
measured in degrees, minutes and seconds of arc as referenced to
"Theta" and only a measurement using x-ray diffraction is accurate
enough to measure angles of the required precision. The actual
cutting process is a mechanical process and tool accuracy/precision,
wear, cutting rate, etc. all have a bearing on the outcome of the
process. Therefore, tolerances come into play and the outcome is a
"range" of angles" centered on a nominal. This translates into a
group of quartz plates with a "range" of Frequency vs. Temperature
characteristics centered around a nominal. Ideally all the plates
will meet the desired Frequency vs. Temperature specification. In
the case of Doubly rotated AT crystals, DRATs, the angle has to be
controlled around two crystallographic axes which is an order of
magnitude more challenging then that of a standard AT Cut device.

Adding to the difficulty of maintaining accuracy of the cut
angle/stability external mechanical attachments and associated
stresses also affect a crystal's temperature characteristics. In
crystal design, the phenomenon is known as Angle Rotation and
crystal designers utilize techniques to minimize the effect of Angle
Rotation.